Question

Find a first order differential equation for the given family of curves. Circles through (-1,0) and (1,0)

Short answer

Answer: y' = -\frac{x}{y-h}

Step-by-step solution

Find the General Equation of the Circles

Since all the circles pass through the points (-1,0) and (1,0), we know that the distance between the major axis points is 2. This means that the center of any given circle must lie on the line x=0. Let the center of any circle be (0,h), where h is the y-coordinate of the center, and r be the radius of the circle. Then, the distance from the center (0, h) to one of the points (-1, 0) is equal to the radius r. We can write this relationship: r^2 = ((-1^2) + (0 - h)^2) Next, we express the general equation of the circles in the family, which takes the form: (x - 0)^2 + (y - h)^2 = r^2

Find the First Order Differential Equation

Now we'll find the first order differential equation, which is the derivative of the equation with respect to x. Differentiating implicitly: 2(x - 0)(1) + 2(y - h)(\frac{dy}{dx}) = 0 Now let's solve for dy/dx (y'): 2x + 2(y - h)(y') = 0 y' = -\frac{2x}{2(y - h)} = -\frac{x}{y-h} We now have the first order differential equation: y' = -\frac{x}{y-h} This differential equation represents the family of circles that pass through the points (-1, 0) and (1, 0).

Key concepts

Implicit Differentiation

Implicit differentiation is a technique in calculus used to find the derivative of a function when it is not explicitly solved for one variable in terms of another. For example, the equation of a circle doesn't solve for y in terms of x or vice versa. When we come across such a situation, we can't differentiate directly as we would for explicit functions. Instead, we use implicit differentiation to take the derivative of both sides of an equation with respect to x.

Using this technique, each term involving y must be differentiated as if y were a function of x, which means applying the chain rule. That's exactly what we did in the solution given. After differentiating, you get an expression involving both y and the derivative of y with respect to x. This is useful for finding slopes of tangent lines or for solving differential equations that describe families of curves, like the one stated in our problem.

Family of Curves

The term 'family of curves' refers to a set of curves that are related by one or more parameters. In the context of differential equations, these parameters often appear as constants that can be varied. Different values of the parameters yield different curves within the family. For instance, in circle equations, the parameter might be the radius or the coordinates of the center. Varying these parameters can shift or resize the circle, but every curve in the family will still share certain characteristics—like being centered at a certain line or passing through specific points.

In the exercise, the family of curves is composed of all the circles that pass through the points (-1,0) and (1,0). Here, the variable 'h' in the circle's equation is the parameter that changes, leading to different y-coordinates of the centers of the circles, each representing a different member of the family of curves. The first order differential equation derived from the family's general equation then encapsulates all these possible circles.

Circle Equations

The equation of a circle in the coordinate plane is typically expressed as \( (x - a)^2 + (y - b)^2 = r^2 \), where \( (a, b) \) is the center of the circle and \( r \) is the radius. However, when we look at a family of circles that goes through specific points, we have additional information that helps us define the equation more narrowly.

In our given exercise, every circle passes through \( (-1, 0) \) and \( (1, 0) \) which implies symmetry about the y-axis, necessitating that the x-coordinate of the center of each circle is zero. This insight simplifies our circle's equation to \( x^2 + (y - h)^2 = r^2 \), where \( h \) is the y-coordinate of the center and can vary. Remember, the varying parameter \( h \) makes this equation represent not just one circle, but an entire family—each with a different center point on the y-axis. Using implicit differentiation to find the slope \( \frac{dy}{dx} \) enables us to explore the properties of tangents to circles within this family.